Spectral systems: new instances and algorithms

The classification problem in algebraic topology motivates the development of increasingly refined invariants capable of distinguishing complex spaces. Spectral sequences have long played a central role in this endeavor, providing structured approximations to homological and homotopical invariants. More recently, spectral systems, introduced by Benjamin Matschke, have emerged as a broad generalization of spectral sequences, allowing filtrations over more general indexing objects and offering greater flexibility in relating graded data to homology. In this thesis we investigate spectral systems both from a theoretical and a computational perspective. First of all, we construct and study a spectral system that combines Serre and Eilenberg--Moore spectral sequences, clarifying their interactions and identifying within this unified framework some preludes that were introduced by Neumann and Szymik. This demonstrates that spectral systems provide a natural setting for combining distinct spectral sequences and for revealing structural relations that are not visible at the level of classical constructions. To investigate a general source of spectral systems, we also develop a generalization of multicomplexes to higher-dimensional settings. Several possible notions of generalized multicomplexes are introduced and analyzed, together with their associated spectral systems. We study their behavior under standard algebraic operations and establish conditions under which effective homology can be obtained. These constructions are illustrated through explicit examples, such as towers of twisted Cartesian products and the effective homology of a bicomplex. Finally, we dualize Matschke's geometric approach for the Eilenberg--Moore generalized spectral sequence, providing a study of the homology of multi-factored fibered products of topological spaces. We address the coproduct problem for the Cobar construction, and we define an iterated Cobar for higher dimensions using generalized multicomplexes. In addition, we define a generalized filtration on this new Cobar chain complex that gives the Eilenberg--Moore spectral system. For this new spectral system, as well as for the previous ones, we present algorithms for their computation using the method of effective homology. This method, created by Rubio and Sergeraert, is a tool to compute the homology of complicated and possibly infinite spaces. Moreover, for the two spectral systems that we introduced first, we also present an implementation on the Kenzo system using effective homology techniques. With this, we extend the capabilities of the existing modules of Kenzo and provide practical tools for analysis and future research.

Grantor dc:publisher

Universidad de La Rioja (España)

Year dc:date

2026

Author dc:creator

• Miguel Treviño, Daniel

Contributors dc:contributor

• Romero Ibáñez, Ana (null)
• Guidolin, Andrea (null)

Statement dc:rights

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Language dc:language

eng

Repository record dc:identifier

https://dialnet.unirioja.es/servlet/oaites?codigo=402493

OAI identifier oai:identifier

oai:dialnet.unirioja.es:TES0000023239